Posted by: Shuanglin Shao | February 17, 2009

A continuation in reading: heat flow monotonicity of Strichartz norms

It has been a long time that I haven’t updated this blog. I promised to post my readings on the Bennett-Bez-Carbery-Hundertmark’s short but beautiful paper,  “heat flow monotonicity of Strichartz norm”. Here it goes!

In this paper, the authors applied the method of heat-flow in the setting of  Strichartz inequality for the Schr\”odinger equation and amusingly obtained, among other things, the Strichartz norm

\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)}

is nondecreasing as the initial datum f evolvs under a under quadratic flow when d=1, 2. This immediately yields that the Gaussians are extremisers to the classical Strichartz inequality,
\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)} \le S_d\|f\|_{L^2(\bf{R}^d)}, \text{ with } S_d:=\frac {\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)}}{\|f\|_{ L^2(\bf{R}^d)}}.
The paper also consider the monotonicity given by some other flows such as Mehler-flow; it also considers the higher dimensions’s analogues by embedding the usual Strichartz norm into one-parameter family of norms \||\cdot\||_p. In this post I will focus on the monotonicity induced by the heat-flow.

The method of heat-flow deformation is interesting in itself. It is expected to be applied to related problems such as the existence of extremisers to the adjoint Fourier restriction operators to the sphere; but it is a different story; so far I can see an immediate difficulty in the latter setting, i.e., an representation formula for \|\widehat{fd\mu}\|_{L^{2+\frac 4d}_x(\bf{R}^{d+1})}, where d\mu is the standard surface measure for the unit sphere in \bf R^{d+1}.

Fortunately the representation formula of this kind is available in the paraboloid setting thanks to Hundertmark-Zharnitsky’s work; I already reported it in my previous post but I would like to record it here again basically because an “equation” is rare in analysis: for nonnegative f\in L^2(\bf R),
\|e^{is\Delta}f\|^6_{L^6_{s,x}(\bf R\times \bf R)}=\frac {1}{\sqrt{12}}\int_{\bf R^3} (f\bigotimes f\bigotimes f) (X)P_1( f\bigotimes f\bigotimes f)(X) dX,
where P_1: L^2(\bf{R}^3)\to L^2(\bf{R}^3) is the projection operator onto the subspace of functions on \bf R^3 which are invariant under the isometries which fix the direction (1,1,1). ( We have a similiar representation formula when d=2. )

The paper innovatively combines this formula with monotonicity given by some heat-flow version of Cauchy-Schwartz inequality together to prove the following main theorem.

Let f\in L^2(\bf{R}^d). If (p,q,d) is Schr\”odinger admissible and q is an even integer which divides p then the one-parameter family of functions

Q_{p,q}(t):= \|e^{is\Delta}(e^{t\Delta}|f|^2)^{1/2}\|_{ L^p_sL^q_x(\mathbf{R}\times \mathbf{R}^d)}

is nondecreasing for all t>0; i.e., Q_{p,q}(t) is nondecreasing in the case (1,6,6), (1,8,4) and (2,4,4). Next we show how to prove this theorem in the case (1,6,6).

{\bf  1. \text{ Monotonicity of } \Lambda(t)}
Given n\in \mathcal{N} and nonnegative integrable functions f_1, f_2 on \bf R^n, we define \Lambda(t):=\int_{\bf R^n} (e^{t\Delta}f_1)^{1/2}(e^{t\Delta}f_2)^{1/2}. Then \Lambda(t) is nondecreasing for all t>0. Instead of repeating the proof in the paper of using the convolution with the heat-kernel, we will try working out how it is deduced from the divergence theorem. Our goal is to prove the following explicit formula
\Lambda'(t)=\frac 14 \int_{\bf R^n} |\nabla (\log e^{t\Delta}f_1)-\nabla (\log e^{t\Delta}f_2)|^2 (e^{t\Delta}f_1)^{1/2}(e^{t\Delta}f_2)^{1/2}.

Let U_j:=e^{t\Delta }f_j=\frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} e^{-\frac {|x-y|^2}{4t}}f_j(y)dy for j=1,2. Then
\partial_t U_j=-\frac {n}{2t} U_j +\frac {1}{(4\pi t)^{n/2}}\int_{\bf R^n}e^{-\frac {|x-y|^2}{4t}}\frac {|x-y|^2}{4t^2}f_j(y)dy.

We write U_j= \frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} e^{-\frac {|y|^2}{4t}}f_j(x-y)dy. Then
\frac {d^2}{d^2x_i}U_j=\frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} (\frac {y_i^2}{4t^2}-\frac {1}{2t})e^{-\frac {|y|^2}{4t}}f_j(x-y)dy. Then
\Delta U_j= \frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n}(\frac {|y|^2}{4t^2} -\frac {n}{2t})e^{-\frac {|y|^2}{4t}}f_j(x-y)dy.
Hence
\partial_t U_j = \Delta U_j. This is not surprising because U_j formally solves the heat equation.

Let V_j:= \nabla(\log U_j ), then
\partial_t (\log U_j) =\frac {(U_j)_t}{U_j}=\frac {\Delta U_j}{U_j}=div (V_j)+|V_j|^2.

Hence we see
\Lambda'(t) =\frac 12 \int_{\bf R^n} [\partial_t (\log V_1)+\partial_t \log V_2] \sqrt{U_1U_2}
\qquad =\frac 12 \int_{\bf R^n} [div (V_1)+div (V_2)+|V_1|^2+|V_2|^2]  \sqrt{U_1U_2}
\qquad =\frac 14 \int_{\bf R^n } |V_1-V_2|^2  \sqrt{U_1U_2}.
Here we have used,
div(\sqrt(U_1U_2) V_1)=\sum_{i=1}^n (V_1)_i \sqrt{U_1U_2}+ (\frac {V_1}{2}+\frac {V_2}{2})\sqrt{U_1U_2}V_1, which yields
0=\int_{\bf R^n} div(\sqrt{U_1U_2} V_1)+div(\sqrt{U_1U_2} V_2)
\quad \int_{\bf R^n } \left(div (V_1)+div (V_2)+\frac {|V_1|^2}{2}+\frac {|V_2|^2}{2}+V_1V_2 \right) \sqrt{U_1U_2},
i.e.,
\int_{\bf R^n } \left(div (V_1)+div (V_2)\right) \sqrt{U_1U_2}=-\int_{\bf R^n}\left(\frac {|V_1|^2}{2}+\frac {|V_2|^2}{2}+V_1V_2\right)\sqrt{U_1U_2}.

{\bf 2. \text{ Monotonicity of } Q_{6,6}(t) }

We start with a general representation formula for the projection map in L^2. For functions in G\in L^2(\bf R^3) we write
P_1 G(X) =\int_O G(\rho X) d\mathcal{H}(\rho),
where O is the group of isometries on \bf R^3 which fix the direction (1,1,1), and d\mathcal{H}(\rho) denotes the right-invariant Haar probability measure on O.

Let F:=f\bigotimes f\bigotimes f and F_\rho:=F(\rho \cdot). Then by using the fact the heat-flow operator commutes with tensor product and Hundertmark-Zharnistsky representation formula, we have
Q_{6,6}^6=\frac {1}{\sqrt{12}} \int_O \int_{\bf R^3} (e^{t\Delta}|F|^2)^{1/2}(X) (e^{t\Delta}|F_\rho|^2)^{1/2}(X)dXd\mathcal{H}(\rho).
This is in form of \Lambda(t). Then the monotonicity of Q_{6,6} follows from that of \Lambda(t) and the non-negativity of the measure d\mathcal{H}.

{\bf 3. \text{ Gaussians are extremisers }}

For f\in L^2, define u(t,x)=H_t*|f|^2(x) with H_t:=\frac {1}{(4\pi t)^{d/2}}e^{-\frac {|x|^2}{4t}}, and the rescaled
\tilde{u}(t,x) =t^{-d} u(t^{-2},t^{-1}x)=\frac {1}{(4\pi)^{d/2}}\int_{\bf R^d} e^{-\frac 14 |x-tv|^2}|f(v)|^2dv.
Then
Q_{6,6}(t^{-2}) =\|e^{is\Delta} (\tilde{u}(t,\cdot))^{1/2}\|_{L^6_{s,x}}.
This leads to
\lim_{t\to \infty} Q_{6,6}(t)=\|e^{is\Delta}(H_1^{1/2})\|_{L^6_{s,x}} \|f\|_{L^2}.
On the other hand,
\lim_{t\to 0} Q_{6,6}(t)=\|e^{is\Delta}|f|\|_{L^6_{s,x}}.
We observe that, by Plancherel theorem in both variables,
\|e^{is\Delta}f\|^3_{L^6_{t,x}}=\|(fd\sigma)^\vee(fd\sigma)^\vee(fd\sigma)^\vee\|_{L^2_{s,x}}
\qquad = \|(fd\sigma)*(fd\sigma)*(fd\sigma)\|_{L^2_{\tau,\xi}}
\qquad \le \|(|f|d\sigma)*(|f|d\sigma)*(|f|d\sigma)\|_{L^2_{\tau,\xi}}=Q^3_{6,6},
where d\sigma denotes the surface measure of the paraboloid in \bf R^2. Hence we obtain
\|e^{is\Delta}f\|_{L^6_{t,x}}\le \|e^{is\Delta}(H_1^{1/2})\|_{L^6_{s,x}} \|f\|_{L^2}.
This shows that H_1^{1/2} is an extremiser. So we are done.

{\bf 4. \text{ A final word}}
You can see that the heat-flow deformation method is a very nice method; it has been applied by the authors in several settings such as the Hausdorff-Young inequality, and Young’s inequality (but I haven’t carefully read their previous works). It is also proved effective in treating d-linear analogues of the Strichartz estimate (Bennett-Carbery-Tao’s work) and in the setting of multilinear Brascamp_Lieb inequalities (Bennett-Carbery-Christ-Tao) (I admitted that I haven’t read them carefully either). The “disadvantage” is that this method doesn’t charaterize the set of extremisers. Of course, we can’t hope it is a tool of catch-all. The question of charaterization is a completely different problem.

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